Every fiberwise principal \(G\) -bundle \(\pi \colon P\rightarrow M\)  is a
fiberwise fiber bundle in the sense that each point of \(M\)  has an open neighborhood \(U\)  such that
the restriction of \(P\)  to \(U\)  is isomorphic to the trivial fiberwise \(G\)  bundle \(U\times _B G\) .
Such a trivial fiberwise fiber bundle is clearly an \(f\) -fibration in the sense of
Theorem REF . When \(B\)  is a point it is a well known theorem
that every numerable fiber bundle \(E\rightarrow M\)  is a Hurewicz fibration. There is an obvious
extension of this notion to the notion of a numerable fiberwise fiber bundle: a fiberwise
fiber bundle is numerable if it is fiberwise locally trivial relative to a numerable open cover
of the base space. We have the following theorem
from [1]}.
